The representation type of rational normal scrolls
Let \((X,\mathcal{O}_X(1))\) be a polarized projective variety of dimension \(n\). A sheaf \(E\) on \(X\) is called Arithmetically Cohen-Macaulay (ACM), if all its middle cohomologies vanish. That is, \(H^i(X,E(k))=0\) for all \(k\) and \(0<i<n\), where as usual, one writes \(E(k)\) to mean \(E\otimes\mathcal{O}_X^{\otimes k}\). For such an \(E\), one can see easily that the number of generators of \(\bigoplus H^0(X,E(k))\) is at most \(\deg X\cdot \mathrm{rk}\, E\). An \(E\) where equality holds are called Ulrich bundles and they have been studied intensely and introduced by \textit{B. Ulrich} [Math. Z. 188, 23--32 (1984; Zbl 0573.13013)]. The paper under review shows that on most rational normal scrolls there are arbitrarily large rank and dimension families of indecomposable Ulrich bundles. The exceptions, a small number, were previously known to have not many indecomposable Ulrich bundles [\textit{R.-O. Buchweitz, G.-M. Greuel} and \textit{F.-O. Schreyer}, Invent. Math. 88, 165--182 (1987; Zbl 0617.14034)].
scientific article; zbMATH DE number 6149393
scientific article